In the first stage of development
(cf. section )
we will
keep \(N=1\)  and
simplify and reformulate
the very concept of the Hermitization.
For this purpose
we will just review the two
complementary existing forms of the theory.
In its older version
(cf. review [1]} or subsection REF  below) one
manages to circumvent
the (in general, difficult) description of the
direct change \(H\rightarrow \mathfrak {h}\)  of the Hamiltonian.
This is mediated by
introduction of another, third Hilbert
space \({\cal H}_{phys}\)  [2]}.
In subsection REF  we will briefly review
the newer and better known
specific approach to the technicalities
known as \({\cal PT}-\) symmetric
quantum mechanics [3]}.
The practical feasibility of the
Hermitization process is
enhanced there simply by a judicious
factorization of the
inner-product metric
(cf. formula (REF ) below).
Naturally,
the ultimate amended physical space \({\cal H}_{phys}\)  must be
constructed as equivalent to its “user-unfriendly”
textbook predecessor \({\cal L}\) .
By construction [3]}, [5]}, Hilbert space \({\cal H}_{phys}\) 
only differs from manifestly unphysical \({\cal H}_{math}\) 
by an amendment and factorization of the inner-product metric
(i.e., in our present notation, of operator \(\Theta \) )
in a way illustrated by the
diagram of Eq. (REF ) below.
